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Square matrix
About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.
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TL;DR: In this article, a second-order statistics-based algorithm is proposed for two-dimensional (2D) direction-of-arrival (DOA) estimation of coherent signals, which is solved by arranging the elements of the correlation matrix of the signal received from a uniform rectangular array to a block Hankel matrix.
Abstract: A second-order statistics-based algorithm is proposed for two-dimensional (2D) direction-of-arrival (DOA) estimation of coherent signals. The problem is solved by arranging the elements of the correlation matrix of the signal received from a uniform rectangular array to a block Hankel matrix. In noiseless cases, it is shown that the rank of the block Hankel matrix equals the number of the DOAs and is independent of the coherency of the incoming waves. Therefore, the signal subspace of the block Hankel matrix can be estimated properly and spans the same column space as the array response matrix. Two matrix pencil pairs containing the DOA parameters are extracted from the signal subspace. This matrix pencil-based estimation problem is then resolved using our previously proposed pairing-free 2D parameter estimation algorithm. Simulation results show that the proposed algorithm outperforms the spatial smoothing method in terms of mean square error (MSE).
56 citations
TL;DR: An extension of the QZ algorithm, called the combination shift $QZ$ algorithm, is presented for solving the generalized matrix eigenvalue problem $Ax = \lambda Bx$ with real square matrices A and B.
Abstract: An extension of the $QZ$ algorithm, called the combination shift $QZ$ algorithm, is presented for solving the generalized matrix eigenvalue problem $Ax = \lambda Bx$ with real square matrices A and B. Features and properties of the various $QR$-type algorithms which were not practical or possible to implement in the $QZ$ algorithm are generalized and implemented in this new algorithm. The emergence of infinite eigenvalues under exact arithmetic is discussed in detail. Results of numerous test cases are presented to give practical application tests and comparisons for the algorithm.
56 citations
TL;DR: It is shown how the matrix square root is related to the constant block coefficient of the inverse of a suitable matrix Laurent polynomial, which allows one to design an efficient algorithm for its computation.
Abstract: We give a new characterization of the matrix square root and a new algorithm for its computation. We show how the matrix square root is related to the constant block coefficient of the inverse of a suitable matrix Laurent polynomial. This fact, besides giving a new interpretation of the matrix square root, allows one to design an efficient algorithm for its computation. The algorithm, which is mathematically equivalent to Newton's method, is quadratically convergent and numerically insensitive to the ill-conditioning of the original matrix and works also in the special case where the original matrix is singular and has a square root.
55 citations
TL;DR: The Frobenius eigenvector of a positive square matrix is obtained by iterating the multiplication of an arbitrary positive vector by the matrix as mentioned in this paper, and the speed of convergence increases statistically with the dimension of the matrix.
Abstract: The Frobenius eigenvector of a positive square matrix is obtained by iterating the multiplication of an arbitrary positive vector by the matrix. Brody (1997) noticed that, when the entries of the matrix are independently and identically distributed, the speed of convergence increases statistically with the dimension of the matrix. As the speed depends on the ratio between the subdominant and the dominant eigenvalues, Brody's conjecture amounts to stating that this ratio tends to zero when the dimension tends to infinity. The paper provides a simple proof of the result. Some mathematical and economic aspects of the problem are discussed.
55 citations
01 Sep 1996
TL;DR: In this article, the authors present results for a quadratic eigenproblem that they encountered in solving an acoustics problem, specifically in modeling the propagation of waves in a room in which one wall was constructed of sound absorbing material.
Abstract: High-dimensional eigenproblems often arise in the solution of scientific problems involving stability or wave modeling. In this article we present results for a quadratic eigenproblem that we encountered in solving an acoustics problem, specifically in modeling the propagation of waves in a room in which one wall was constructed of sound-absorbing material. Efficient algorithms are known for the standard linear eigenproblem, Ax = x
where A is a real or complex-valued square matrix of order n. Generalized eigenproblems of the form Ax = Bx, which occur in nite element formulations, are usually reduced to the standard problem, in a form such as B Ax = x. The reduction requires an expensive inversion operation for one of the matrices involved. Higher-order polynomial eigenproblems are also usually transformed into standard eigenproblems. We discuss here the second-degree (i.e., quadratic) eigenproblem 2C2 + C1 + C0 x = 0 in which the matrices Ci are square matrices.
55 citations